A few questions about proofs regarding the Binomial Coefficient. [closed]

Question

I have two things that I need to prove using Combinatorial proofs (where you ask a question that satisfies both sides of an equation) and I have no idea what I'm doing.

1. Prove $C(n,3) = C(2,2) + C(3,2) + C(4,2) + \ldots + C(n-1,2)$. The question that I need to be answered is, "How many 3 element subsets does an n element set have?" The first part is trivial, as the answer is $nC3$ by the definition of the Binomial Coefficient. However, I have no idea how to prove the second half of the equation satisfies the question. That is, how would I go about proving an $n$ element set has $C(2,2) + C(3,2) + C(4,2) + \ldots + C(n-1,2)$ subsets of 3 elements?

2. Prove $k C(n,k) = n C(n-1,k-1)$ combinatorial proof. While this is trivial to prove algebraically, I have to find a question that both sides of the equation answer, similar to the problem above with the 3 element subsets.

3. Finally, does anybody have any tips for getting through this class? The median score on our first exam was ~58% and our professor doesn't give us a curve. I got a 55, sitting at around a D- with the few homework grades we've gotten back. I am really pretty decent at other math classes, but this class has been kicking my proverbial ass. I'm just venting right here, but still. I need help.

Answer 1

HINTS:

1. Classify the $3$-element subsets of $\{1,2,\ldots,n\}$ according to their largest elements. If $3\le k\le n$, how many $3$-element subsets of $\{1,2,\ldots,n\}$ have $k$ as largest element?

2. You have a pool of $n$ players. How many ways are there to pick a team of $k$ of those players, assigning one of the chosen players to be captain? If you still don’t see it after giving it some serious thought, take a look at the further hint in the spoiler-protected block below.

You can choose the team and then the captain, or you can choose the captain and then the rest of the team.